Finite Group Databasebeta

← #10576531

Subgroups — #10576531

Order 512.

6236 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62860C211yesmaximal, characteristic
2#62849C211yesmaximal, characteristic, complemented
3#61344C2116yesmaximal, complemented
4#61342C2116yesmaximal, complemented
5#61335C218yesmaximal, complemented
6#61328C218yesmaximal, complemented
7#60620C212yesmaximal, complemented
8#60612C211yesmaximal, characteristic
9#60491C211yesmaximal, characteristic, complemented
10#60489C212yesmaximal, complemented
11#60424C211yesmaximal, characteristic
12C2×#3044C212yesmaximal, complemented
13#60193C211yesmaximal, characteristic, complemented
14#60169C212yesmaximal, complemented
15C2×#2950C211yesmaximal, characteristic, complemented
16C22×#137C2212yes—
17C23×Q16C2211yescharacteristic
18#3510C2218yes—
19#3509C2218yes—
20#3507C2218yes—
21#3502—24no—
22#3502—28no—
23#3502C2218yescomplemented
24#3502C22116yescomplemented
25#3495—28no—
26#3491—24no—
27#3461—24no—
28#3458—24no—
29#3458C2212yes—
30#3458C2211yescharacteristic
31#3458C2211yescharacteristic
32#3449C2211yescharacteristic
33C2×#536C2212yes—
34#3428C2212yes—
35C2×#519C2212yes—
36C2×#519C2211yescharacteristic
37C2×#519C2211yescharacteristic
38#3377C22132yescomplemented
39#3377C22132yescomplemented
40#3375C22116yescomplemented
41#3374C22116yescomplemented
42#3374C22132yescomplemented
43#3371C22132yescomplemented
44#3369C22132yescomplemented
45#3365C22116yescomplemented
46#3364C22132yescomplemented
47#3363C22116yescomplemented
48#3140C2218yescomplemented
49#3139C2218yescomplemented
50#3137C2218yes—
51#3137C22116yes—
52#3137C22116yescomplemented
53#3136C2218yes—
54#3135C2214yes—
55#3131C2214yes—
56#3084C2218yescomplemented
57#3083C2218yescomplemented
58#3082C2218yescomplemented
59#3082C22116yescomplemented
60#3081C2218yescomplemented
61#3078C2218yescomplemented
62#3076C22116yes—
63#3044C22116yes—
64#3044C2218yes—
65#3044C2218yescomplemented
66#3044C2218yescomplemented
67#3043C2218yes—
68#3037C2218yes—
69C2×#464C2218yescomplemented
70C2×#462C2218yescomplemented
71#2962C22116yescomplemented
72#2961C2214yescomplemented
73#2960C2214yescomplemented
74#2950C22116yescomplemented
75#2950C2218yescomplemented
76#2949C2218yescomplemented
77#2947C22116yescomplemented
78C4×#137C22116yes—
79C2×#441C2214yescomplemented
80C2×C4×Q16C2214yescomplemented
81#2911C2212yes—
82C2×#427C2211yescharacteristic
83#2905C2211yescharacteristic
84C2×#424C2212yes—
85#2896C2212yes—
86#2896C2212yes—
87C2×#419C2212yes—
88C2×#419C2211yescharacteristic
89C2×#419C2211yescharacteristic
90C2×#416C2212yes—
91#2873C2212yes—
92C2×#405C2211yescharacteristic
93C22×#41C2312yes—
94C23×Q8C2311yescharacteristic
95C23×Q8C2311yescharacteristic
96C2×#137—216no—
97C2×#137C2318yes—
98C2×#137C2318yes—
99C22×Q16—28no—
100C22×Q16C2312yes—
101C22×Q16C2312yes—
102C22×Q16C2314yes—
103C22×Q16C2318yes—
104C22×SD16C2314yes—
105C22×M16C2311yescharacteristic
106C22×M16C2312yes—
107C23×C8C2311yescharacteristic
108#559C23116yes—
109Q82—216no—
110Q82—28no—
111Q82C23116yes—
112#557—216no—
113#557—28no—
114#557—216no—
115#557—216no—
116#557—216no—
117#557C23116yes—
118#557C23132yes—
119#557C23116yes—
120#556—28no—
121#556C23132yes—
122#554—28no—
123#552—216no—
124#552—232no—
125#551—232no—
126#544—216no—
127#544—232no—
128#544—22no—
129#544—24no—
130#544D414yes—
131#544D418yes—
132#544C2318yes—
133#544C2318yes—
134#544C2318yes—
135#543—22no—
136#543—24no—
137#543D414yes—
138#542—216no—
139#541—216no—
140#539—216no—
141#537—22no—
142#536—24no—
143#536—24no—
144#536C2318yes—
145#536C2314yes—
146#536C2318yes—
147#535C2314yes—
148#534—22no—
149#533—216no—
150#533—216no—
151#533—232no—
152#533—24no—
153#533—24no—
154#533—24no—
155#533C2318yes—
156#533C2314yes—
157#533C2314yes—
158#533C2318yes—
159#532C2314yes—
160C2×C4⋊Q8C2314yes—
161#529—216no—
162#529—216no—
163C2×#124C2314yes—
164#525—24no—
165C2×#123—24no—
166C2×C22⋊Q8—24no—
167#520—24no—
168#520C2318yes—
169#519—24no—
170#519—24no—
171#519—24no—
172#519—24no—
173#519C2318yes—
174#519C2314yes—
175#519C2318yes—
176#519C2318yes—
177#519C2318yes—
178#518—24no—
179#518C2314yes—
180C4×#41—24no—
181C4×#41C2318yes—
182C2×C4×Q8—22no—
183C2×C4×Q8—22no—
184C2×C4×Q8—24no—
185C2×C4×Q8C2314yes—
186C2×C4×Q8C2314yes—
187C2×C4×Q8C2318yes—
188C2×#117—24no—
189C2×#117—22no—
190C2×#117—24no—
191C2×#117—22no—
192C2×#117C2312yes—
193C2×#117C2311yescharacteristic
194C2×#117C2311yescharacteristic
195C2×#117C2311yescharacteristic
196C2×#117C2312yes—
197#491C2318yes—
198#491C23132yes—
199#490C2318yes—
200#489C23116yes—
201#489C23132yes—
202#488C23116yes—
203#488C23132yes—
204#487C2318yes—
205#487C23132yes—
206#486C2318yes—
207#479C23116yes—
208#479C23116yes—
209#479C23132yes—
210#478C23116yes—
211#478C23116yes—
212#477C23116yes—
213#477C23116yes—
214#476C23116yes—
215#475C23116yes—
216#475C23116yes—
217#474C23116yes—
218#464C23132yes—
219#464C23132yes—
220#464C23132yes—
221#462C23132yes—
222#462C23116yes—
223#462C23132yes—
224#460C23116yes—
225#446C23132yes—
226#446C23116yes—
227C8×Q8C23116yes—
228#441C23132yes—
229#441C23132yes—
230#440C23132yes—
231C4×Q16C23132yes—
232C4×Q16C23116yes—
233C4×Q16C23116yes—
234C4×SD16C23132yes—
235#428C2318yes—
236#428C2318yes—
237#427C2318yes—
238#427C2318yes—
239C2×#107C2312yes—
240C2×#106C2312yes—
241#424C2318yes—
242#424C2318yes—
243#424C2314yes—
244#424C2314yes—
245#423C2314yes—
246C2×C4⋊C8C2318yes—
247#419C2318yes—
248#419C2314yes—
249#419C2318yes—
250#419C2314yes—
251#419C2318yes—
252#419C2318yes—
253#418C2314yes—
254#418C2314yes—
255#417C2318yes—
256#417C2318yes—
257#416C2318yes—
258#416C2318yes—
259#416C2314yes—
260#416C2314yes—
261#416C2318yes—
262C2×#103C2314yes—
263C2×#103C2318yes—
264C2×#103C2314yes—
265C2×#103C2314yes—
266C2×#102C2314yes—
267#405C2318yes—
268#405C2318yes—
269#405C2318yes—
270C4×M16C2318yes—
271C2×#97C2312yes—
272C2×C4×C8C2312yes—
273C2×#41—22no—
274C2×#41—22no—
275C2×#41—28no—
276C2×#41—28no—
277C2×#41C2414yes—
278C2×#41C2414yes—
279C22×Q8—21no—
280C22×Q8—22no—
281C22×Q8—21no—
282C22×Q8—22no—
283C22×Q8—24no—
284C22×Q8—24no—
285C22×Q8—24no—
286C22×Q8#4118yes—
287C22×Q8C2412yes—
288C22×Q8C2414yes—
289C22×Q8C2412yes—
290C22×Q8C2414yes—
291C22×Q8C2412yes—
292C22×D4—22no—
293C22×D4C2412yes—
294C23×C4—21no—
295C23×C4—21no—
296C23×C4—22no—
297C23×C4C2411yescharacteristic
298#137—232no—
299#137—232no—
300C2×Q16—232no—
301C2×Q16—28no—
302C2×Q16—28no—
303C2×Q16—216no—
304C2×Q16C2418yes—
305C2×Q16C2418yes—
306C2×Q16C24116yes—
307C2×SD16—216no—
308C2×SD16C24116yes—
309C2×M16—28no—
310C2×M16—24no—
311C2×M16C2414yes—
312C2×M16C2414yes—
313C2×M16C2414yes—
314C22×C8—24no—
315C22×C8C2411yescharacteristic
316C22×C8C2411yescharacteristic
317C22×C8C2414yes—
318C22×C8C2414yes—
319C22×C8C2412yes—
320C4⋊Q8—232no—
321C4⋊Q8—216no—
322C4⋊Q8—232no—
323C4⋊Q8—216no—
324C4⋊Q8—28no—
325C4⋊Q8—28no—
326C4⋊Q8—28no—
327C4⋊Q8—28no—
328C4⋊Q8C2×D4116yes—
329C4⋊Q8C2×D418yes—
330C4⋊Q8C24116yes—
331C4⋊Q8C2414yes—
332C4⋊Q8C24116yes—
333C4⋊Q8C2418yes—
334#127C2414yes—
335#126—232no—
336#126—232no—
337#125—216no—
338#125—232no—
339#125—232no—
340#125—232no—
341#125—232no—
342#125—28no—
343#125—28no—
344#125—28no—
345#125C2×D4116yes—
346#125C2×D418yes—
347#125C24116yes—
348#125C2418yes—
349#124—216no—
350#124—28no—
351#124C24116yes—
352#124C2418yes—
353#123—232no—
354#123—216no—
355#123—216no—
356#123—28no—
357C22⋊Q8—232no—
358C22⋊Q8—28no—
359C22⋊Q8—216no—
360C22⋊Q8—216no—
361C22⋊Q8—216no—
362C22⋊Q8—28no—
363C22⋊Q8—216no—
364C22⋊Q8—216no—
365C22⋊Q8—216no—
366C22⋊Q8C2×D4116yes—
367C22⋊Q8C2×D4132yes—
368#121—28no—
369#121—28no—
370#120—28no—
371C4×Q8—216no—
372C4×Q8—216no—
373C4×Q8—216no—
374C4×Q8—216no—
375C4×Q8—216no—
376C4×Q8—216no—
377C4×Q8—28no—
378C4×Q8—232no—
379C4×Q8—216no—
380C4×Q8—28no—
381C4×Q8—216no—
382C4×Q8—28no—
383C4×Q8—216no—
384C4×Q8—28no—
385C4×Q8C24116yes—
386C4×Q8C24132yes—
387C4×Q8C24116yes—
388C4×Q8C24116yes—
389C4×Q8C24116yes—
390C4×D4—216no—
391C4×D4—28no—
392C4×D4—216no—
393C4×D4—216no—
394C4×D4—28no—
395C4×D4C24116yes—
396#117—22no—
397#117—22no—
398#117—22no—
399#117—22no—
400#117—28no—
401#117—232no—
402#117—28no—
403#117—216no—
404#117—216no—
405#117—216no—
406#117—216no—
407#117—28no—
408#117C2×D414yes—
409#117C2×D414yes—
410#117C2×D414yes—
411#117C2414yes—
412#117C2414yes—
413#117C2414yes—
414#117C2414yes—
415#117C2414yes—
416#117C2414yes—
417#117C2414yes—
418C2×C4⋊C4—24no—
419C2×C4⋊C4—24no—
420C2×C4⋊C4—28no—
421C2×C4⋊C4—28no—
422C2×C4⋊C4—28no—
423C2×C4⋊C4C2×D412yes—
424C2×C4⋊C4C2×D414yes—
425C2×C4⋊C4C2412yes—
426C2×C4⋊C4C2414yes—
427C2×C4⋊C4C2412yes—
428C2×C22⋊C4—21no—
429C2×C22⋊C4—22no—
430C2×C22⋊C4—21no—
431C2×C22⋊C4—28no—
432C2×C22⋊C4—28no—
433C2×C22⋊C4C2×D412yes—
434C2×C22⋊C4C2×D414yes—
435C2×C42—24no—
436C2×C42—24no—
437C2×C42—28no—
438C2×C42—21no—
439C2×C42—21no—
440C2×C42—22no—
441C2×C42—28no—
442C2×C42C2412yes—
443C2×C42C2414yes—
444#107C2414yes—
445#107C2414yes—
446#107C24116yes—
447#106C2418yes—
448#106C24116yes—
449C4⋊C8C24116yes—
450C4⋊C8C24116yes—
451C4⋊C8C24116yes—
452#103C2418yes—
453#103C2418yes—
454#103C24116yes—
455#103C24116yes—
456#103C24132yes—
457#103C24116yes—
458#103C24116yes—
459#102C2418yes—
460#102C2418yes—
461#102C24116yes—
462#102C24116yes—
463#97C2418yes—
464#97C24116yes—
465C4×C8C2418yes—
466C4×C8C24116yes—
467C24—22no—
468C24C22×D411yescharacteristic
469#41—28no—
470#41—28no—
471#41—28no—
472#41—28no—
473#41—48no—
474#41—48no—
475C2×Q8—216no—
476C2×Q8—28no—
477C2×Q8—24no—
478C2×Q8—22no—
479C2×Q8—22no—
480C2×Q8—24no—
481C2×Q8—22no—
482C2×Q8—28no—
483C2×Q8—216no—
484C2×Q8—24no—
485C2×Q8—28no—
486C2×Q8—28no—
487C2×Q8—24no—
488C2×Q8—24no—
489C2×Q8—24no—
490C2×Q8—28no—
491C2×Q8—24no—
492C2×Q8—48no—
493C2×Q8—44no—
494C2×Q8C22×D4116yes—
495C2×Q8C2×#41116yes—
496C2×Q8C2×#4118yes—
497C2×Q8C2514yes—
498C2×Q8C2514yes—
499C2×Q8C2514yes—
500C2×D4—24no—
501C2×D4—24no—
502C2×D4—22no—
503C2×D4—28no—
504C2×D4—24no—
505C2×D4—24no—
506C2×D4—48no—
507C2×D4C2514yes—
508C22×C4—28no—
509C22×C4—24no—
510C22×C4—22no—
511C22×C4—22no—
512C22×C4—24no—
513C22×C4—22no—
514C22×C4—22no—
515C22×C4—24no—
516C22×C4—22no—
517C22×C4—24no—
518C22×C4—24no—
519C22×C4—216no—
520C22×C4—44no—
521C22×C4—48no—
522C22×C4C22×D414yes—
523C22×C4C22×D418yes—
524C22×C4C22×D411yescharacteristic
525C22×C4C22×D412yes—
526C22×C4C2×#4114yes—
527C22×C42−1+414yes—
528C22×C4C2511yescharacteristic
529C22×C4C2512yes—
530Q16—216no—
531Q16—216no—
532Q16—232no—
533SD16—232no—
534M16—28no—
535M16—28no—
536M16—28no—
537C2×C8—28no—
538C2×C8—22no—
539C2×C8—22no—
540C2×C8—28no—
541C2×C8—24no—
542C2×C8C2514yes—
543C2×C8C2512yes—
544C2×C8C2518yes—
545C2×C8C2512yes—
546C4⋊C4—22no—
547C4⋊C4—216no—
548C4⋊C4—216no—
549C4⋊C4—216no—
550C4⋊C4—232no—
551C4⋊C4—232no—
552C4⋊C4—24no—
553C4⋊C4—22no—
554C4⋊C4—28no—
555C4⋊C4—28no—
556C4⋊C4—216no—
557C4⋊C4—216no—
558C4⋊C4—216no—
559C4⋊C4—216no—
560C4⋊C4C22×D418yes—
561C4⋊C4C22×D418yes—
562C4⋊C4C22×D418yes—
563C4⋊C4C22×D4132yes—
564C4⋊C4C22×D4116yes—
565C4⋊C4C2518yes—
566C4⋊C4C2514yes—
567C4⋊C4C2514yes—
568C22⋊C4—24no—
569C22⋊C4—24no—
570C22⋊C4—24no—
571C22⋊C4—24no—
572C22⋊C4—216no—
573C22⋊C4—232no—
574C22⋊C4—216no—
575C22⋊C4C22×D418yes—
576C22⋊C4C22×D418yes—
577C22⋊C4C22×D418yes—
578C42—24no—
579C42—216no—
580C42—216no—
581C42—216no—
582C42—232no—
583C42—24no—
584C42—24no—
585C42—24no—
586C42C22×D418yes—
587C42C22×D414yes—
588C42C2518yes—
589C42C2514yes—
590C23—24no—
591C23—42no—
592C23—44no—
593C23—84no—
594C23#54814yes—
595C23C23×D412yes—
596C23C23×D411yescharacteristic
597Q8—24no—
598Q8—28no—
599Q8—216no—
600Q8—24no—
601Q8—24no—
602Q8—24no—
603Q8—24no—
604Q8—416no—
605Q8—44no—
606Q8—48no—
607Q8—44no—
608Q8—816no—
609Q8—88no—
610Q8C22×#4118yes—
611D4—24no—
612D4—24no—
613D4—48no—
614D4—48no—
615D4—44no—
616D4—88no—
617C2×C4—24no—
618C2×C4—216no—
619C2×C4—28no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—28no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—48no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—48no—
629C2×C4—44no—
630C2×C4—48no—
631C2×C4—48no—
632C2×C4—816no—
633C2×C4—82no—
634C2×C4—82no—
635C2×C4—84no—
636C2×C4#54818yes—
637C2×C4#54814yes—
638C2×C4C23×D418yes—
639C2×C4C23×D4116yes—
640C2×C4C23×D412yes—
641C2×C4C23×D411yescharacteristic
642C2×C4C22×#4114yes—
643C2×C4C22×#4112yes—
644C2×C4C2×2−1+412yes—
645C2×C4C2×2−1+414yes—
646C2×C4C2611yescharacteristic
647C8—22no—
648C8—22no—
649C8—28no—
650C8—24no—
651C22—22no—
652C22—24no—
653C22—44no—
654C22—48no—
655C22—44no—
656C22—164no—
657C22C2×#54814yes—
658C22C2×#54812yes—
659C22C24×D411yescharacteristic
660C4—216no—
661C4—28no—
662C4—28no—
663C4—44no—
664C4—44no—
665C4—48no—
666C4—44no—
667C4—44no—
668C4C2×#54812yes—
669C4C2×#54814yes—
670C4C23×#4111yescharacteristic
671C4C22×2−1+411yescharacteristic
672C2—24no—
673C2—22no—
674C2—84no—
675C2C22×#54811yescharacteristic
676C2#6309312yes—
subgroupG/H
center, socleC22C24×D4
commutator, Frattini, cosocleC2×C4C26
prime radical#3449C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ #3458 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #62849C2semidirect
3. #61344C2semidirect
4. #61342C2semidirect
5. #61335C2semidirect
6. #61328C2semidirect
7. #60620C2semidirect
9. #60491C2semidirect
10. #60489C2semidirect
12. C2×#3044C2semidirect
13. #60193C2semidirect
14. #60169C2semidirect
15. C2×#2950C2semidirect
23. #3502C22semidirect
24. #3502C22semidirect
38. #3377C22semidirect
39. #3377C22semidirect
40. #3375C22semidirect
41. #3374C22semidirect
42. #3374C22semidirect
43. #3371C22semidirect
44. #3369C22semidirect
45. #3365C22semidirect
46. #3364C22semidirect
47. #3363C22semidirect
48. #3140C22semidirect
49. #3139C22semidirect
52. #3137C22semidirect
56. #3084C22semidirect
57. #3083C22semidirect
58. #3082C22semidirect
59. #3082C22semidirect
60. #3081C22semidirect
61. #3078C22semidirect
65. #3044C22semidirect
66. #3044C22semidirect
69. C2×#464C22semidirect
70. C2×#462C22semidirect
71. #2962C22semidirect
72. #2961C22semidirect
73. #2960C22semidirect
74. #2950C22semidirect
75. #2950C22semidirect
76. #2949C22semidirect
77. #2947C22semidirect
79. C2×#441C22semidirect
80. C2×C4×Q16C22semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#628602
#628492
#613442
#613422
#613352
#613282
#606202
#606122
#604912
#604892
#604242
C2×#30442
#601932
#601692
C2×#29502